I'll give an overview of recent progress in the structure and classification of simple amenable C*-algebras, making parallels to the Connes-Haagerup classification of amenable von Neumann algebras and drawing examples from group actions.
Tag - Von Neumann algebras
It is shown that the operator space generated by peripheral eigenvectors of a unital completely positive map on a von Neumann algebra has a C*-algebra structure. This extends the notion of non-commutative Poisson boundary by including the point spectrum of the map contained in the unit circle. The main ingredient is dilation theory. This theory provides a simple formula for the new product. The notion has implications to our understanding of quantum dynamics. For instance, it is shown that the peripheral Poisson boundary remains invariant in discrete quantum dynamics.
In recent joint work with Yidong Chen, we discovered spectral gap estimates and concentration inequalities for for dynamics with few generators. Some of these estimates are dimension free and then can be used to feed in the recent theory of complexity initiated by Lloyd and Jaffe, and adapted more recently for specific resources. The goal is to find a viable theory of complexity which holds in type II1 and III1 von Neumann algebras, both of which come naturally in quantum field theory and Witten's take on black holes.
Functional inequalities are potent tools in analysing the convergence time of quantum Markov semigroups. Specifically, the modified log-Sobolev inequality (MLSI) concerns the (exponential) convergence of the time evolution in terms of relative entropy as a quantitative measure. In this talk, I will present an estimate of modified log-Sobolev constant using completely positive mixing time. Our proof uses only entropic inequalities, which gives a unified information-theoretic approach that applies in both classical and quantum setting, even the Type III von Neumann algebras. For a quantum analogue of birth-and-death process, our estimate is tight up to a factor of absolute constant. As an application, I will talk about the use of (complete) modified log-Sobolev constant in estimating the decay of relative entropy of entanglement.
An online lecture course by the University of Münster in Von Neumann algebras.
Lecture 1: We'll briskly review basic properties of semi-finite von Neumann algebras. The standard representation, completely positive maps, group von Neumann algebras, the group-measure space construction, and some characterizations of the hyperfinite II1 factor.
Lecture 2: We discuss some approximation properties that are common in "rank 1" groups: Weak amenability and biexactness.
Lecture 3: We discuss properly proximal groups as defined by Boutonnet, Ioana, and myself, and give some applications to group von Neumann algebras associated to higher-rank groups.
Lecture 4: We’ll introduce measure equivalence (ME), W*-equivalence (W*E), and von Neumann equivalence (VNE). We’ll give examples and discuss invariants.
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